Error bound for hybrid models of two

Comments

Transcription

Error bound for hybrid models of two
Error bound for hybrid models of two-scaled
stochastic reaction systems
Tobias Jahnke and Vikram Sunkara
1 Introduction
Biological systems such as gene-regulatory networks and cell metabolic processes
consist of multiple species which are undergoing transformations via a set of reaction channels. If all populations are sufficiently large, then the evolution of the
concentrations over time can be modeled by the classical reaction-rate equation, i.e.
a system of ordinary differential equations; cf. [23]. In many applications, however,
some of the species occur in low amounts, and it was observed that small stochastic fluctuations in their populations can cascade large effects to the other species.
Important examples are gene-regulatory networks where the evolution of the entire system depends crucially on the stochastic behavior of a rather small number
of transcription factors. In order to capture these effects, such systems must be described by a Markov jump processes, which respects the inherent discrete nature of
the system and its stochastic interactions.
The associated time-dependent probability distribution is the solution of the
Chemical Master Equation (CME), but solving the CME is a considerable challenge, as the size of the state space scales exponentially in the number of species
(curse of dimensionality). For this reason, Monte Carlo approaches based on the
stochastic simulation algorithm from [6] or related methods are often used. In an
alternative line of research, numerical techniques have been applied to the CME in
order to reduce the number of degrees of freedom, e.g. optimal state space truncation [1, 26, 27], spectral approximation [4], adaptive wavelet compression [16, 20],
sparse grids [11], or tensor product approximation [2, 10, 18, 22, 21] among othTobias Jahnke
Karlsruhe Institute of Technology, Department of Mathematics, Kaiserstr. 93, D-76133 Karlsruhe,
Germany, e-mail: [email protected]
Vikram Sunkara
Karlsruhe Institute of Technology, Department of Mathematics, Kaiserstr. 93, D-76133 Karlsruhe,
Germany, e-mail: [email protected]
1
2
Tobias Jahnke and Vikram Sunkara
ers. But in spite of the progress achieved with these approaches, many biological
systems are still out of reach of direct numerical approximation.
The size of the problem can be significantly reduced if only species with low populations are described by a probability distribution, whereas the abundant species are
represented by (conditional) moments. This approach is motivated by the famous result in [23] which states, roughly speaking, that stochastic fluctuations in large populations are insignificant. In the last years, this has inspired the development of hybrid
models where a low-dimensional CME is coupled to ordinary differential equations
similar to the classical reaction-rate equation; cf. [5, 9, 11, 12, 13, 17, 25, 28].
In this article, we analyze the accuracy of a hybrid model called MRCE (model
reduction based on conditional expectations). This approach has been proposed in
[9, 17, 25], and it was demonstrated numerically that MRCE captures the critical
bi-modal solution profiles which appear in certain applications. In [9, 17, 28], numerical techniques for MRCE were introduced, and an error bound for the modeling
error was proven in [28]. In the present article, we make the additional assumption
that the reaction system involves two scales, i.e. that the ratio between the small
and large populations is proportional to a scaling parameter 0 < ε 1. For such
two-scaled systems, we prove that the modeling error of the MRCE approximation
is proportional to ε. The proof blends ideas and techniques from [19] and [28].
2 The Chemical Master Equation of two-scale reaction systems
We consider a partitioned reaction system with two groups of species denoted
by S1 , . . . , Sd and Sd+1 , . . . , Sd+D , respectively, with d, D ∈ N. Let X(t) ∈ Nd0 be
the vector whose entries X1 (t), . . . Xd (t) indicate how many copies of each of the
species S1 , . . . , Sd exist at time t ∈ [0,tend ], and let Y (t) = (Y1 (t), . . . ,YD (t)) contain
the copy numbers of Sd+1 , . . . , Sd+D . The species interact via r ∈ N reaction channels , R1 , . . . , Rr , each of which is represented by a scheme
d
Rj :
D
cj
∑ a jk Sk + ∑ b jk Sd+k −→
k=1
k=1
d
D
∑ abjk Sk + ∑ bb jk Sd+k ,
k=1
(1)
k=1
with a jk , abjk , b jk , b
b jk ∈ N0 and c j > 0. If the j-th reaction channel fires, then the
population numbers jump from the current state (X(t),Y (t)) = (n, m) ∈ Nd0 × ND
0 to
the new state (n, m) + (ν j , µ j ), where (ν j , µ j ) ∈ Zd+D is the stoichiometric vector
associated to R j , i.e.
T
ν j = abj1 − a j1 , . . . , abjd − a jd ∈ Zd
T
µj = b
b j1 − b j1 , . . . , b
b jd − b jd ∈ ZD .
Error bound for hybrid models of two-scaled stochastic reaction systems
3
In stochastic reaction kinetics, the function t 7→ (X(t),Y (t)) is a realization of a
Markov jump process; cf. [6, 14]. According to [6] the transition rates of this process
depend on the propensity functions of the reaction channels. We assume that the
propensity function of R j has the form α j (n)β j (m) with
d
α j (n) = c j ∏
k=1
nk
,
a jk
β j (m) = ε
γ( j)−1 |b j |
ε
D
∏
k=1
mk
,
b jk
(2)
where |b j | = ∑di=1 b ji , and where 0 < ε 1 is a scaling parameter discussed below.
The value of γ depends on whether or not the population numbers of the first group
of species change when R j fires. To be more precise, we partition the index set
{1, . . . , r} into
J1 = {1, . . . , r} \ J0
J0 = j ∈ {1, . . . , r} : ν j = (0, . . . , 0)T ,
and let γ be the indicator function
γ( j) =
0 if j ∈ J0 ,
1 if j ∈ J1 .
(3)
The reason for this particular scaling is the following: if (X(t),Y (t)) = (n, m) ∈ Nd0 ×
−1 , then α (n)β (m) = O(c ε γ( j)−1 ) for all j =
ND
j
j
j
0 with n ∈ O(1) and m ∈ O ε
1, . . . , r. Hence,
the
population
numbers
of
S
,
.
.
.
,
S
may
change with a rate
d+1
d+D
of O ε −1 , whereas the populations of S1 , . . . , Sd only change with a rate of O(1),
provided that c j = O(1) for all j. For initial data X(0) = O(1) and
Y (0) = O ε −1 ,
one can thus expect that E(X(t)) = O(1) and E(Y (t)) = O ε −1 on bounded time
intervals. Hence, ε is roughly speaking the ratio between the small and the large
population numbers of the two groups S1 , . . . , Sd and Sd+1 , . . . , Sd+D , respectively.
This scaling was extensively motivated and illustrated in [19], and a very similar
scaling was considered in [25]. For d = 0 and α j (n) = c j , our scaling coincides
with the thermodynamic limit which has been analyzed in [23]. For ε = 1, there is
no qualitative difference between the two groups of species, such that our setting
corresponds to the situation in [6] where no partition nor scaling is considered.
Let p(t, n, m) be the probability that at time t ≥ 0 the process is in state (n, m) ∈
Nd0 × ND
0 , i.e. the probability that there are nk copies of Sk for k = 1, . . . , d, and ml
copies of Sd+l for l = 1, . . . , D. It is well-known (see [6, 7]) that the probability
distribution p evolves according to the Chemical Master Equation (CME)
r
∂t p(t, n, m) =
(4)
∑ α j (n − ν j )β j (m − µ j )p(t, n − ν j , m − µ j )
j=1
− α j (n)β j (m)p(t, n, m)
∀(n, m) ∈ Nd0 × ND
0
p(0, n, m) = p0 (n, m)
(5)
4
Tobias Jahnke and Vikram Sunkara
with the convention that p(t, n − ν j , m − µ j ) = 0 if n − ν j 6∈ Nd0 or m − µ j 6∈ ND
0 . For
the sake of a more compact notation we define the shift operators S1j and S2j by
S1j u(n, m) =
S2j u(n, m) =
u(n − ν j , m)
0
if n − ν j ∈ Nd0
else
u(n, m − µ j )
0
if m − µ j ∈ ND
0
else
for u : Nd0 × ND
0 −→ R. The two shift operators commute, i.e.
u(n − ν j , m − µ j ) if n − ν j ∈ Nd0 , m − µ j ∈ ND
1 2
2 1
0
S j S j u(n, m) = S j S j u(n, m) =
0
else.
Products of functions are to be understood entry-wise, and applying a shift operator
to a product u(n, m)v(n, m) is to be understood in the sense that
S1j uv (n, m) = S1j (uv) (n, m) = u(n − ν j , m)v(n − ν j , m) = S1j u S1j v (n, m).
With these operators, the CME (4) can be reformulated as
r
∂t p =
∑ (S1j S2j − I) (α j β j p) .
(6)
j=1
The chemical master equation (6) is considered on the space
n
o
`1 = u : Nd0 × ND
0 −→ R : ∑ ∑ |u(n, m)| < ∞ .
n∈Nd0 m∈ND
0
of absolutely summable functions on Nd0 × ND
0 . This is a straightforward extension
of the standard `1 -space. For vector-valued functions u = (u1 , . . . , uN ) : Nd0 ×ND
0 −→
RN with some N > 1, u ∈ `1 means that u j ∈ `1 for all j = 1, . . . , N. The space `1 is
endowed with the norm
kuk`1 =
∑ ∑
|u(n, m)|
n∈Nd0 m∈ND
0
where | · | = | · |1 is the 1-norm on RN . We set X 0 = `1 and define the spaces X i
via the recursion
n
o
X i+1 = u ∈ X i | (n, m) 7→ mk u(n, m) ∈ X i for all k ∈ {1, . . . , D} .
If p(t, ·, ·) ∈ `1 is the solution of the CME (6), then p1 (t, n) = ∑m p(t, n, m) is the
marginal distribution of p(t, ·, ·), and if p1 (t, n) 6= 0, then
p2 (t, m | n) =
p(t, n, m)
p1 (t, n)
(7)
Error bound for hybrid models of two-scaled stochastic reaction systems
5
is the conditional probability that at time t there are m j particles of S j given there
are ni particles of Si (i ∈ {1, . . . , d}, j ∈ {d + 1, . . . , d + D}). If p(t, ·, ·) ∈ X 2 , then
the conditional central moments
ξ (t, n) =
∑
mp2 (t, m | n)
(8)
m∈ND
0
Cξ (t, n) =
∑D
T
m − ξ (t, n) m − ξ (t, n) p2 (t, m | n)
m∈N0
exist provided that p1 (t, n) 6= 0.
3 Model reduction based on conditional expectations
Solving the CME (4) or (6) numerically is a considerable challenge. First, the infinite state space Nd0 × ND
0 has to be truncated; this causes an error which has been
analyzed in [26]. The truncated state space is finite, but still (d + D)-dimensional,
and the total number of states is usually so large that standard numerical schemes
cannot be applied.
On the other hand, the solution of the CME often provides more information
than actually needed to understand the biological process. In many applications,
one is mainly interested in the question how the stochastic behavior of S1 , . . . , Sd affects the dynamics of Sd+1 , . . . , Sd+D . If the population numbers of Sd+1 , . . . , Sd+D
are sufficiently large, then stochastic fluctuations within their populations can be
neglected according to [23]. In this case, it is sufficient to compute the marginal distribution p1 (t, n) of the species S1 , . . . , Sd along with conditional moments which
measure the abundance of Sd+1 , . . . , Sd+D . This has motivated the construction of
hybrid models: Instead of trying to solve the high-dimensional CME and then extracting the relevant information from the solution p(t, n, m), one derives a reduced
set of equations, namely a low-dimensional CME for the marginal distribution coupled with other ODEs; cf. [5, 9, 11, 12, 13, 17, 25, 28]. Hybrid models have the
advantage that the huge number of unknowns is significantly reduced, which makes
the problem computationally feasible. The price to pay is that hybrid models involve
structurally more complicated differential equations than the (linear) CME, and that
such a model reduction causes a modeling error in addition to the numerical error.
The following hybrid model has been derived in [17]:
∂t w = ∑ (S1j − I) α j β j (φ )w =: A(φ )w
(9)
j∈J1
∂t (φ w) =
r
α j β j (φ )φ w + ∑ µ j S1j α j β j (φ )w
∑ (S1j − I)
j∈J1
=: F(φ , w) + G(φ , w)
j=1
(10)
6
Tobias Jahnke and Vikram Sunkara
For fixed φ , (9) is again a CME, but on the lower-dimensional state space Nd0 . The
function w(t, n) approximates the marginal distribution p1 (t, n) of the full CME solution, whereas φ (t, n) approximates the conditional expectations ξ (t, n) defined in
(8). This is why this model was called model reduction based on conditional expectations (MRCE) in [17]. It was demonstrated by numerical examples that MRCE
captures certain bimodal solution profiles correctly, in contrast to simpler hybrid
models proposed in the literature. Since w and φ do not depend on m any more, the
(d + D)-dimensional state space of the CME is replaced by a d-dimensional state
space, which reduces the computational costs considerably. Similar approaches have
been proposed in [8, 9, 13, 15, 24, 25] for the CME and related differential equations.
Approximating the conditional expectations ξ (t, n) has the drawback that (7)
and hence (8) cannot be properly defined if p1 (t, n) = 0. The same applies to the
approximations w(t, n) ≈ p1 (t, n) and φ (t, n) ≈ ξ (t, n). The hybrid model (9)-(10)
is formulated in terms of w and φ w, but in order to evaluate the term β j (φ ) on the
right-hand side, we have to divide φ (t, n)w(t, n) by w(t, n). This is only possible if
w(t, n) > 0, and for w(t, n) ≈ 0 such a division causes numerical instability. Different
strategies to cope with this problem have been proposed in [9, 13, 17, 25, 28]. Since
the main goal of the present article is an analysis of the accuracy of MRCE, we will
avoid such technical problems by the following assumption:
Assumption 1 We assume that the CME (6) with initial condition (5) has a unique
classical solution p(t, ·, ·) ∈ `1 with strictly positive marginal distribution p1 (t, ·),
i.e. p1 (t, n) > 0 for all t ∈ [0,tend ] and all n ∈ Nd0 . This implies that p2 , ξ and Cξ are
well-defined. Moreover, we assume that the hybrid model (9)-(10) with initial data
w(0, n) = p1 (0, n)
and
φ (0, n) = ξ (0, n)
(11)
has a unique solution, and that w(t, ·) ∈ `1 is strictly positive for all t ∈ [0,tend ].
This assumption seems to be a strong simplification because in typical applications it
can be observed that for every threshold parameter δ ∈ (0, 1), there are only finitely
many states with p(t, n, m) ≥ δ . Roughly speaking, this means that p(t, n, m) ≈ 0
for “most of” the states. However, if p(t? , n? , m? ) = 0 for some t? > 0, then the state
(n? , m? ) ∈ Nd0 × ND
0 cannot be reached from neither of the states which had nonzero
probability at time t = 0. As a consequence, one could simply exclude (n? , m? ) from
the state space to avoid the problem, and in this sense, Assumption 1 is not a severe
restriction. Since numerical methods for solving (9)-(10) are not discussed in this
article, numerical instabilities are not an issue here.
4 Error analysis for the hybrid model
Since β j (m) defined in (2) depends on the scaling parameter ε and since β j (m)
appears both in the CME (6) and in the hybrid model (9)-(10), the functions
Error bound for hybrid models of two-scaled stochastic reaction systems
7
p(t, n, m), p1 (t, n), ξ (t, n), w(t, n), and φ (t, n) all depend1 on ε, too. In this section we prove that the modeling error of MRCE is bounded by Cε (see Theorem 1
below). Throughout the article, C denotes a generic constant which may have different values at different occurrences. The proof combines the arguments from [17, 19]
with the analysis from [28] where systems with no scaling have been investigated.
Our error analysis is based on the following assumptions.
Assumption 2 For every j ∈ {1, . . . , r} we assume that |b j | ≤ 2.
This is a natural assumption, because the probability of a trimolecular reaction is
negligible according to [7, page 418].
Assumption 3 We assume that the solution of the CME (6) satisfies p(t, ·, ·) ∈ X 3
for t ∈ [0,tend ] and that
(n, m) 7→ α j (n)p(t, n, m) ∈ X 3
for all j ∈ {1, . . . , r}.
Assumption 4 We assume that
sup sup |ξ (t, n)| ≤
t∈[0,tend ] n∈Nd
0
C
,
ε
sup sup |Cξ (t, n)| ≤
t∈[0,tend ] n∈Nd
0
C
,
ε
C
sup sup |φ (t, n)| ≤
ε
t∈[0,tend ] n∈Nd
0
with a constant which does not depend on ε. Moreover, we assume that all third
central moments of p2 (t, · | n) are bounded by Cε −2 with a constant which does not
depend on t ∈ [0,tend ], ε, and n ∈ Nd0 .
Assumption 5 Suppose that there is a constant C > 0 such that for all t ∈ [0,tend ]
and j ∈ {1, . . . , r} the bound
max α j (·)u(t, ·)`1 ≤ C ku(t, ·)k`1
j=1,...,r
holds for each of the following functions:
u = p1 ,
u = β j ξ p1 − β j φ w,
u = β j (ξ )ξ p1 − β j (φ )φ w.
Note that Assumption 4 implies u ∈ `1 in each case.
The following error bound for the modeling error of MRCE is the main result of this
article.
Theorem 1. Under the assumptions 1, 2, 3, 4, and 5, there is a constant Cb > 0 such
that the approximation error of MRCE is bounded by
1
We do not make this dependency explicit in the notation in order to keep the equations as simple
as possible.
8
Tobias Jahnke and Vikram Sunkara
sup kp1 (t, ·) − w(t, ·)k`1 ≤ Cb ε
(12)
t∈[0,tend ]
sup kξ (t, ·)p1 (t, ·) − φ (t, ·)w(t, ·)k`1 ≤ Cb .
(13)
t∈[0,tend ]
If in addition
|b j | ≤ 1 for all j ∈ J0 ,
|b j | = 0 for all j ∈ J1 ,
(14)
then MRCE is even exact, i.e. we can choose Cb = 0 in (12) and (13).
According to (13) the error of the approximation ξ p1 ≈ φ w remains bounded, but
does not decrease when ε → 0. This is not obvious, because Assumption
4 implies
that kξ (t, ·)p1 (t, ·)k`1 = O ε −1 and kφ (t, ·)w(t, ·)k`1 = O ε −1 . Multiplying both
sides of (13) by ε shows that the relative error converges linearly in ε.
Proof. It will be shown below in Lemma 2 and Lemma 3 that
kp1 (t, ·) − w1 (t, ·)k`1 + ε kξ p1 (t, ·) − φ w(t, ·)k`1
≤ Cb ε +C
Zt
ε k(ξ p1 − φ w) (s, ·)k`1 ds +C
0
Zt
(15)
kp1 (s, ·) − w(s, ·)k`1 ds.
0
for all t ∈ [0,tend ] with constants Cb and C which do not depend on t or ε. Hence, the
Gronwall lemma yields
kp1 (t, ·) − w1 (t, ·)k`1 + ε kξ p1 (t, ·) − φ w(t, ·)k`1 ≤ Cb ε
which proves (12) and (13). Moreover, it will be shown that we can choose Cb = 0
in the special case (14).
t
u
The remainder of this article is devoted to the proof of the Gronwall inequality (15).
As a preparatory step, we prove the following lemma:
Lemma 1. Let y : Nd0 −→ Rd , z : Nd0 −→ Rd with
max |y(n)| ≤ C/ε,
n∈Nd0
max |z(n)| ≤ C/ε,
n∈Nd0
(16)
and let u ∈ `1 and v ∈ `1 . Then for every j ∈ {1, . . . , r}, there is a constant C > 0
such that
β j (y)u − β j (z)v 1 ≤ Cε γ( j) kyu − zvk 1 + ε −1 ku − vk 1
`
`
`
with γ( j) defined in (3). Note that the assumption (16) implies that yu − zv ∈ `1 .
A similar lemma has been shown in [19, Lemma 4].
Proof. For |λ j | = 0 the assertion is obvious, because in this case β j (y) = ε γ( j)−1 is
Error bound for hybrid models of two-scaled stochastic reaction systems
9
constant. If |λ j | = 1, then there is a k ∈ {1, . . . , d} such that β j (y) = ε γ( j) yk , and the
assertion follows. If |λ j | = 2, then the propensity β j (y) takes the form
c if k 6= l
β j (y) = ĉ j ε γ( j)+1 yk yl
with ĉ j = 1 j
2 c j if k = l
for some k, l ∈ {1, . . . , d}. Thus, we have to bound the difference
β j (y)u − β j (z)v = ĉ j ε γ( j)+1 (yk yl u − zk zl v)
!
= ĉ j ε
γ( j)+1
yk (yl u − zl v) + yk zl (v − u) + zl (yk u − zk v) .
Since (16) implies that |εyk (n)| ≤ C and |εzl (n)| ≤ C, it follows that
β j (y)u − β j (z)v 1 ≤ Cε γ( j) kyu − zvk 1 +Cε γ( j)−1 ku − vk 1
`
`
`
t
u
which proves the assertion.
Lemma 2. Under the assumptions of Theorem 1 there are constants Cb ≥ 0 and
C > 0 such that
kp1 (t, ·) − w1 (t, ·)k`1 ≤ Cb ε + C
Zt
ε k(ξ p1 − φ w) (s, ·)k`1 ds
0
Zt
+C
kp1 (s, ·) − w(s, ·)k`1 ds.
0
for all t ∈ [0,tend ]. The constants Cb and C do not depend on t or ε, and we can
choose Cb = 0 in the special case (14).
Proof. From the definition of the marginal distribution p1 it follows that
∂t p1 =
(S1j S2j − I)α j β j p +
∑ ∑
j∈J0 m∈ND
(S1j S2j − I)α j β j p.
∑ ∑
j∈J1 m∈ND
0
0
The first sum vanishes, because S1j = I for j ∈ J0 , and
∑D (S2j − I)α j β j p = 0
(17)
m∈N0
by Lemma 2 in [19]. Since S1j S2j − I = S1j (S2j − I) + (S1j − I) and since (17) implies
∑ ∑D S1j (S2j − I)α j β j p = 0,
j∈J1 m∈N
0
we obtain
10
Tobias Jahnke and Vikram Sunkara
∂t p1 =
(S1j − I)α j β j p =
∑ ∑
∑ (S1j − I)α j ∑D β j p2 p1 .
j∈J1
j∈J1 m∈ND
0
(18)
m∈N0
Following the ideas of [3] we use the Taylor expansion
1
β j (m) = β j (ξ ) + ∇β j (ξ )T (m − ξ ) + (m − ξ )T ∇2 β j (m − ξ )
2
(19)
where ξ = ξ (t, n). Since β j is at most quadratic by Assumption 2, all higher-order
terms vanish. This yields2
∑
β j (m)p2 (t, m|n) = β j (ξ ) + R j (t, n)
(20)
m∈ND
0
with R j (t, n) = trace Cξ (t, n)∇2 β j
because ∑m∈ND p2 (t, m|n) = 1 and ∑m∈ND (m − ξ (t, n))p2 (t, m|n) = 0. Substituting
0
0
this into (18) gives
∂t p1 =
∑ (S1j − I)α j β j (ξ )p1 + R = A(ξ )p1 + R
(21)
j∈J1
with a rest term R = R(t, n) given by
R=
∑ (S1j − I)α j R j p1 .
j∈J1
Comparing (21) with (9) yields
∂t p1 − ∂t w = A(ξ )p1 − A(φ )w + R
and since p1 (0, ·) = w(0, ·) according to (11), we obtain
kp1 (t, ·) − w(t, ·)k`1 ≤
Z t
0
A ξ (s, ·) p1 (s, ·) − A φ (s, ·) w(s, ·) 1 ds
`
+
Z t
0
R(s, ·) 1 ds.
`
(22a)
(22b)
Our next goal is to derive a bound for the second term (22b). According to Assumption 4 we have
sup sup |Cξ (t, n)| ≤
t∈[0,tend ] n∈Nd
0
C
,
ε
whereas (2) yields
2
The remainder term R j is not to be mixed up with the reaction channel R j in (1).
Error bound for hybrid models of two-scaled stochastic reaction systems
(
0
∇ βj =
ε γ( j)+1
11
if |b j | ≤ 1
if |b j | = 2.
2
By Assumption 2, no other cases have to be considered. Hence, it follows that
sup sup |R j (s, n)| = sup sup trace Cξ (s, n)∇2 β j ≤ Cε γ( j) ,
(23)
s∈[0,tend ] n∈Nd
0
s∈[0,tend ] n∈Nd
0
and Assumption 5 and the fact that γ( j) = 1 for all j ∈ J1 yield the estimate
Z t
0
R(s, ·) 1 ds ≤ Ctε sup sup kα j (s, ·)p1 (s, ·)k 1 ≤ Cbtend ε.
`
`
(24)
s∈[0,t] j∈J1
If |b j | ∈ {0, 1} for all j = 1, . . . , r, then ∇2 β j = 0 and hence R(s, ·)`1 = 0 such
that one can choose Cb = 0 in the special case (14). The first error term (22a) can be
bounded by
A ξ p1 − A φ w 1
` 1
= ∑ (S j − I) β j ξ α j p1 − ∑ (S1j − I) β j φ α j w `1
j∈J1
j∈J1
≤ C max β j ξ α j p1 − β j φ α j w`1
j∈J1
≤ C max β j ξ p1 − β j φ w`1
j∈J1
(25)
due to Assumption 5. Applying Lemma 1 now yields
β j ξ p1 − β j φ w 1 ≤ Cε γ( j) ξ p1 − φ w 1 + ε −1 p1 − w 1 ,
`
`
`
and since the maximum in (25) is only taken over J1 , it follows that
A ξ p1 − A φ w 1 ≤ C ε ξ p1 − φ w 1 + p1 − w 1 .
`
`
`
Substituting (24) and (26) into (22a) and (22b) yields the assertion.
(26)
t
u
Lemma 3. Under the assumptions of Theorem 1 there are constants Cb ≥ 0 and
C > 0 such that
k(ξ p1 − φ w) (t, ·)k`1 ≤ Cb + C
Zt
k(ξ p1 − φ w) (s, ·)k`1 ds
0
+
C
ε
Zt
kp1 (s, ·) − w(s, ·)k`1 ds.
0
for all t ∈ [0,tend ]. The constants Cb and C do not depend on t or ε. If |b j | ∈ {0, 1}
for all j = 1, . . . , r, then we can choose Cb = 0.
12
Tobias Jahnke and Vikram Sunkara
Proof. With similar arguments as in the proof of Lemma 2, it can be shown that
r
∂t (ξ p1 ) (t, n) =
∑ µ j ∑D
j=1
+
S1j α j β j p (t, n, m)
m∈N0
∑ ∑D m
(S1j − 1)α j β j p (t, n, m)
(27)
j∈J1 m∈N
0
(see step 1 in the proof of Lemma 6 in [19] for details). For the first term on the
right-hand side, (20) yields


∑ S1j α j β j p (t, n, m) = α j (n − ν j )  ∑ β j (m)p2 (t, m | n − ν j ) p1 (t, n − ν j )
m∈ND
0
m∈ND
0
= α j (n − ν j ) β j ξ (t, n − ν j ) + R j (t, n − ν j ) p1 (t, n − ν j )
= S1j α j β j (ξ ) + R j p1 (t, n).
Moreover, it follows from (19) that
mβ j (m)p(t, n, m) = β j (ξ )ξ + T j (t, n) p1 (t, n)
∑
m∈ND
0
with ξ = ξ (t, n) and
1
T j (t, n) =Cξ (t, n)∇β j (ξ ) + ξ R j (t, n)
2
1
+ ∑ (m − ξ )(m − ξ )T (∇2 β j )(m − ξ )p2 (t, m | n).
2 m∈ND
0
Substituting into (27) yields
r
∑ µ j S1j
∂t (ξ p1 ) (t, n) =
α j β j (ξ )p1 (t, n)
j=1
+
∑
(S1j − 1)α j β j (ξ )ξ p1 (t, n) + R(t, n)
j∈J1
with defect
r
R(t, n) =
∑ µ j S1j
R j α j p1 (t, n) +
j=1
∑
(S1j − 1)T j α j p1 (t, n).
j∈J1
Comparing this with (10) shows that
∂t (ξ p1 )(t, n) = F(ξ , p1 ) + G(ξ , p1 ) + R.
Error bound for hybrid models of two-scaled stochastic reaction systems
13
We will now prove that kR(t, ·)k`1 ≤ Cb with a constant Cb ≥ 0 which does not
depend on ε nor on t ∈ [0,tend ]. In the special case (14), we have that ∇β j (ξ ) = 0 for
all j ∈ J1 and ∇2 β j (ξ ) = 0 for all
j = 1, . . . , r. This implies R j = 0 for all j, T j = 0
for all j ∈ J1 and hence R(s, ·)`1 = 0 such that one can choose Cb = 0. If (14) is
not true, then according to (23) we know that |R j (s, n)| ≤ C for all j = 1, . . . , r, and
with straightforward calculations and Assumption 5 we obtain the bound
r
∑ µ j S1j R j α j p1 (t, ·) 1 ≤ C max kα j (·)p1 (t, ·)k`1 ≤ C.
`
j=1
j=1,...,r
Concerning the second term in R, Assumption 4 and (2) imply that
sup sup |T j (t, n)| ≤ Cε γ( j)−1 .
t∈[0,tend ] n∈Nd
0
The sum in the second term of R is only taken over j ∈ J1 such that γ( j) = 1. With
Assumption 5, it thus follows that
∑ ∑ (S1j − 1)T j α j p1 (t, ·) 1 ≤ C max kα j (·)p1 (t, ·)k`1 ≤ C,
`
j∈J1 m∈ND
j=1,...,r
0
which proves kR(t, ·)k`1 ≤ Cb . Now the error ξ p1 − φ w can be estimated by
k(ξ p1 )(t, ·) − (φ w)(t, ·)k`1 ≤
≤
Z t
0
Z t
0
k∂t (ξ p1 )(s, ·) − ∂t (φ w)(s, ·)k`1 ds
kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ds
Z t
+
0
(28)
kG(ξ p1 )(s, ·) − G(φ w)(s, ·)k`1 ds +Ct.
It follows from Assumption 5 and Lemma 1 that
kG(ξ , p1 )(s, ·) − G(φ , w)(s, ·)k`1
≤ C max α j β j (ξ )p1 − β j (φ )w (s, ·)
`1
j=1,...,r
≤ C max β j (ξ )p1 − β j (φ )w (s, ·)`1
j=1,...,r
≤ C kξ p1 (s, ·) − φ w(s, ·)k`1 +
C
kp1 (s, ·) − w(s, ·)k`1 .
ε
(29)
A corresponding bound has to be shown for F(ξ p1 ) − F(φ w). Assumption 5 yields
kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ≤ C max α j β j (ξ )ξ p1 (s, ·) − α j β j (φ )φ w(s, ·) 1
j∈J1
`
≤ C max β j (ξ )ξ p1 (s, ·) − β j (φ )φ w(s, ·) 1 .
j∈J1
`
14
Tobias Jahnke and Vikram Sunkara
We decompose the error into three parts:
kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1
≤ C max β j (ξ )[ξ p1 − φ w](s, ·) 1 + β j (ξ )φ [w − p1 ](s, ·) 1
j∈J1
`
`
+ φ [β j (ξ )p1 − β j (φ )w](s, ·) 1 .
`
Since (2) and Assumption 4 imply that for every j ∈ J1
|b |
sup sup β j ξ (t, n) ≤ Cε γ( j)−1 ε |b j | sup sup ξ (t, n) j
t∈[0,tend ] n∈Nd
t∈[0,tend ] n∈Nd
0
0
≤ Cε
and since supn∈Nd |φ (s, n)| ≤
0
C
ε
γ( j)−1
= C,
by Assumption 4, we obtain
kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1
≤ C[ξ p1 − φ w](s, ·) 1
`
C w(s, ·) − p1 (s, ·) 1 + max +
β j (ξ )p1 − β j (φ )w (s, ·) 1 .
`
j∈J1
ε
`
Applying Lemma 1 to the last term yields
max β j (ξ )p1 − β j (φ )w (s, ·) 1 ≤ Cεk ξ p1 − φ w k`1 +Ckp1 − wk`1
j∈J1
`
because γ( j) = 1 for j ∈ J1 . Hence, we have shown the estimate
kF(ξ p1 )(s, ·) − F(φ w)(s, ·)k`1 ≤Cξ p1 (s, ·) − φ w(s, ·)`1
C
+ w(s, ·) − p1 (s, ·)`1
ε
Substituting (29) and (30) into (28) proves the assertion.
(30)
t
u
References
1. Burrage, K., Hegland, M., MacNamara, S., Sidje, R.B.: A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of
biological systems. In: A.N.Langville, W.J.Stewart (eds.) Markov Anniversary Meeting: An
international conference to celebrate the 150th anniversary of the birth of A.A. Markov, pp.
21 – 38. Boson Books (2006)
2. Dolgov, S.V., Khoromskij, B.N.: Tensor-product approach to global time-space-parametric
discretization of chemical master equation. Tech. rep., Max-Planck-Institut für Mathematik in
den Naturwissenschaften (2012)
Error bound for hybrid models of two-scaled stochastic reaction systems
15
3. Engblom, S.: Computing the moments of high dimensional solutions of the master equation.
Appl. Math. Comput. 180(2), 498–515 (2006)
4. Engblom, S.: Spectral approximation of solutions to the chemical master equation. J. Comput.
Appl. Math. 229(1), 208–221 (2009)
5. Ferm, L., Lötstedt, P., Hellander, A.: A hierarchy of approximations of the master equation
scaled by a size parameter. J. Sci. Comput. 34, 127–151 (2008)
6. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of
coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)
7. Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–
425 (1992)
8. Griebel, M., Jager, L.: The BGY3dM model for the approximation of solvent densities. J.
Chem. Phys. 129(17), 174,511–174,525 (2008)
9. Hasenauer, J., Wolf, V., Kazeroonian, A., Theis, F.: Method of conditional moments for the
chemical master equation. Tech. rep., to appear in J. Math. Biol. (2013)
10. Hegland, M., Garcke, J.: On the numerical solution of the chemical master equation with
sums of rank one tensors. In: W. McLean, A.J. Roberts (eds.) Proceedings of the 15th Biennial
Computational Techniques and Applications Conference, CTAC-2010, pp. C628–C643 (2011)
11. Hegland, M., Hellander, A., Lötstedt, P.: Sparse grids and hybrid methods for the chemical
master equation. BIT 48, 265–284 (2008)
12. Hellander, A., Lötstedt, P.: Hybrid method for the chemical master equation. J. Comput. Phys.
227, 100–122 (2007)
13. Henzinger, T., Mateescu, M., Mikeev, L., Wolf, V.: Hybrid numerical solution of the chemical
master equation. In: P. Quaglia (ed.) Proceedings of the 8th International Conference on
Computational Methods in Systems Biology (CMSB10), pp. 55–65. ACM (2010)
14. Higham, D.J.: Modeling and simulating chemical reactions. SIAM Rev. 50(2), 347–368
(2008)
15. Iedema, P.D., Wulkow, M., Hoefsloot, H.C.J.: Modeling molecular weight and degree of
branching distribution of low-density polyethylene. Macromolecules 33, 7173–7184 (2000)
16. Jahnke, T.: An adaptive wavelet method for the chemical master equation. SIAM J. Sci.
Comput. 31(6), 4373–4394 (2010)
17. Jahnke, T.: On reduced models for the chemical master equation. SIAM Multiscale Model.
Simul. 9(4), 1646–1676 (2011)
18. Jahnke, T., Huisinga, W.: A dynamical low-rank approach to the chemical master equation.
Bull. Math. Biol. 70(8), 2283–2302 (2008)
19. Jahnke, T., Kreim, M.: Error bound for piecewise deterministic processes modeling stochastic
reaction systems. SIAM Multiscale Model. Simul. 10(4), 1119–1147 (2012)
20. Jahnke, T., Udrescu, T.: Solving chemical master equations by adaptive wavelet compression.
J. Comput. Phys. 229(16), 5724–5741 (2010)
21. Kazeev, V., Khammash, M., Nip, M., Schwab, C.: Direct solution of the chemical master
equation using quantized tensor trains. Tech. rep., ETH Zurich (2013)
22. Kazeev, V., Schwab, C.: Tensor approximation of stationary distributions of chemical reaction
networks. Tech. rep., ETH Zurich (2013)
23. Kurtz, T.G.: The relationship between stochastic and deterministic models of chemical reactions. J. Chem. Phys. 57, 2976–2978 (1973)
24. Lötstedt, P., Ferm, L.: Dimensional reduction of the Fokker-Planck equation for stochastic
chemical reactions. Multiscale Model. Simul. 5, 593–614 (2006)
25. Menz, S., Latorre, J.C., Schütte, C., Huisinga, W.: Hybrid stochastic-deterministic solution of
the chemical master equation. SIAM Multiscale Model. Simul. 10(4), 1232–1262 (2012)
26. Munsky, B., Khammash, M.: The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124(4), 044,104 (2006)
27. Sunkara, V.: Analysis and numerics of the chemical master equation. Ph.D. thesis, Australian
National University (2013)
28. Sunkara, V.: Finite state projection method for hybrid models. Tech. rep., Karlsruhe Institute
of Technology (2013)

Similar documents